Categories of Dimension Zero

TL;DR

This paper characterizes categories of dimension zero over a ring k by introducing a combinatorial and linear-algebraic 'homological modulus' that determines when finitely generated representations have finite length.

Contribution

It provides a new characterization of dimension zero categories using a homological modulus, linking representation theory with combinatorial and linear algebraic conditions.

Findings

Categories of dimension zero are characterized by the existence of a homological modulus.

Finitely generated representations of such categories have finite length.

The homological modulus offers a practical criterion for identifying dimension zero categories.

Abstract

If D is a category and k is a commutative ring, the functors from D to k-Mod can be thought of as representations of D. By definition, D is dimension zero over k if its finitely generated representations have finite length. We characterize categories of dimension zero in terms of the existence of a "homological modulus" (Definition 1.4) which is combinatorial and linear-algebraic in nature.